Sobolev regularity of polar fractional maximal functions. (September 2020) - González-Riquelme, Cristian
Abstract: We study the Sobolev regularity on the sphere S d of the uncentered fractional Hardy–Littlewood maximal operator M ˜ β at the endpoint p = 1, when acting on polar data. We first prove that if q = d d − β, 0 < β < d and f is a polar W 1, 1 ( S d ) function, we have ‖ ∇ M ˜ β f ‖ q ≲ d, β ‖ ∇ f ‖ 1 . We then prove that the map f ↦ | ∇ M ˜ β f | is continuous from W 1, 1 ( S d ) to L q ( S d ) when restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f ↦ | ∇ M ˜ β f | from W rad 1, 1 ( R d ) to L q ( R d ) . Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy–Littlewood maximal operator M β implies the continuity of the map f ↦ | ∇ M β f | from W 1, 1 to L q, in the context of polar functions on S d and radial functions on R d .